4 circles, externally tangent in pairs

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jlacosta

6 minutes ago

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k a My Retirement & New Leadership at AoPS

rrusczyk 1571

N Mar 26, 2025 by SmartGroot

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rrusczyk

Mar 24, 2025

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Mar 26, 2025

Polynomial

EtacticToe 2

N 23 minutes ago by yuribogomolov

Source: Own

Let $f(x)$ be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer $a,b,c,d$ such that $f(a)=…=f(d)=5$ .

Find all $k$ such that $f(k)=8$

2 replies

EtacticToe

Dec 14, 2024

yuribogomolov

23 minutes ago

Is this FE solvable?

Mathdreams 2

N 29 minutes ago by Mathdreams

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy$ for all reals $x$ and $y$ .

2 replies

+1 w

Mathdreams

Yesterday at 6:58 PM

Mathdreams

29 minutes ago

Prove that there are no tuples $(x, y, z)$ sastifying $x^2+y^2-z^2=xyz-2$

Anabcde 1

N 29 minutes ago by giangtruong13

Prove that there are no tuples $(x, y, z) \in \mathbb{Z}^3$ sastifying $x^2+y^2-z^2=xyz-2$

1 reply

Anabcde

2 hours ago

giangtruong13

29 minutes ago

inequalities hard

Cobedangiu 3

N 29 minutes ago by sqing

problem

.................

3 replies

Cobedangiu

Mar 31, 2025

sqing

29 minutes ago

No more topics!

4 circles, externally tangent in pairs

parmenides51 1

N May 25, 2021 by jampm

Source: Mexican Mathematical Olympiad 2000 OMM P1

Circles $A,B,C,D$ are given on the plane such that circles $A$ and $B$ are externally tangent at $P, B$ and $C$ at $Q, C$ and $D$ at $R$ , and $D$ and $A$ at $S$ . Circles $A$ and $C$ do not meet, and so do not $B$ and $D$ .
(a) Prove that the points $P,Q,R,S$ lie on a circle.
(b) Suppose that $A$ and $C$ have radius $2, B$ and $D$ have radius $3$ , and the distance between the centers of $A$ and $C$ is $6$ . Compute the area of the quadrilateral $PQRS$ .

1 reply

parmenides51

Jul 28, 2018

jampm

May 25, 2021

4 circles, externally tangent in pairs

G H J

Reveal topic tags

geometry

tangent circles

G H BBookmark kLocked kLocked NReply

Source: Mexican Mathematical Olympiad 2000 OMM P1

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parmenides51

30629 posts

parmenides51

#1 h Jul 28, 2018, 10:19 AM • 2 Y

Y by Adventure10, Mango247

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jampm

49 posts

jampm

#2 h May 25, 2021, 10:49 PM

Y by

I've managed to do the first part with spoiler

angle chase

but I've seen a beautiful solution using spoiler

inversion

!
solution

Like let P be the center of inversion! Now A and B will invert into 2 parallel lines since they only intersect at P! Using the information given it is easy to deduce the rest of the inverted picture, basically you just have to show that after inversion Q, R and S are collinear and you'll be done (which is again an easy angle chase but hey, it's a rather beautiful one!).

This post has been edited 1 time. Last edited by jampm, May 25, 2021, 10:50 PM
Reason: better formatting

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